The quadratic function f is negative only on the open interval (−2, 1/4 ) and its graph passes through the point (−1, −5). Find a,b, and c.

you know that the roots are -2 and 1/4.

So, y=a(x+2)(x - 1/2)

Now just use the point given to find a, and then expand if you want the coefficients.

To find the quadratic function, we need to determine the values of a, b, and c in the general form of a quadratic function, which is f(x) = ax^2 + bx + c.

Step 1: Find the value of c.
Since the graph passes through the point (-1, -5), we can substitute x = -1 and f(x) = -5 into the quadratic function to obtain the following equation:
-5 = a(-1)^2 + b(-1) + c
Simplifying this equation, we get: -5 = a - b + c.

Step 2: Determine the sign of a.
We are given that the quadratic function f is negative only on the open interval (-2, 1/4). This means that the parabola opens downwards, indicating a negative leading coefficient (a < 0).

Step 3: Use the information from Step 2 to find the values of a, b, and c.
From Step 2, we know that a < 0.
From Step 1, we have the equation -5 = a - b + c.

Since we have two equations and three variables, we need to make an assumption about one of the variables. Let's assume b = 0.

Substituting b = 0 into the second equation, we get -5 = a + c.

Now we have two equations:
-5 = a - b + c
-5 = a + c

Solving these equations simultaneously:
a - b + c = -5 [Equation 1]
a + c = -5 [Equation 2]

We can subtract Equation 2 from Equation 1 to eliminate the variable c:
(a - b + c) - (a + c) = -5 - (-5)
a - b + c - a - c = 0
-b = 0
b = 0

Substituting b = 0 back into either Equation 1 or Equation 2, we get:
a - 0 + c = -5
a + c = -5

Since a + c = -5, and we have assumed b = 0, we can deduce that a = -5 and c = 0.

Therefore, the values of a, b, and c are:
a = -5
b = 0
c = 0

Hence, the quadratic function f(x) is f(x) = -5x^2.